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Show that int0^(npi+v)|sinx|dx=2n+1-cosv...

Show that `int_0^(npi+v)|sinx|dx=2n+1-cosv ,` where `n` is a positive integer and `,lt=v

Text Solution

Verified by Experts

The correct Answer is:
`2n+1-cosv`
Let `I=int_(0)^(npi+v)|sinx|dx`
`=int_(0)^(v)|sinx|dx+int_(v)^(npi+v)|sinx|dx`
`=int_(0)^(v)sinx dx+n int_(0)^(pi) |sinx+dx` [ `:'|sinx|` has period `pi`]
`=(-cosx)_(0)^(v)+n(-cosx)_(0)^(pi)`
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